A compact extension of Journé's $T1$ theorem on product spaces

Abstract

We prove a compact version of the $T1$ theorem for bi-parameter singular integrals. That is, if a bi-parameter singular integral operator $T$ admits the compact full and partial kernel representations, and satisfies the weak compactness property, the diagonal $\mathrm{CMO}$ condition, and the product $\mathrm{CMO}$ condition, then $T$ can be extended to a compact operator on $L^p(w)$ for all $1<p<\infty$ and $w\in A_p({\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2})$. Even in the unweighted setting, it is the first time to give a compact extension of Journé's $T1$ theorem on product spaces.

References in corpus (2)

• Weighted little bmo and two-weight inequalities for Journé commutators
• Genuinely multilinear weighted estimates for singular integrals in product spaces